Trim balancing of second-order non-linearity in double ended tuning fork resonators

ABSTRACT

A double-ended tuning fork (DETF) sensor having one or more mass balance tabs for equalizing second-order, K2, non-linearity terms between two DETFs, and methods for manufacturing the same.

This application is a Continuation-in-Part of U.S. Utility Ser. No08/873,048 filed in the names of Blake, et al on Jun. 11, 1997 now U.S.Pat. No. 6,282,959, which claims the benefit of U.S. Provisionalapplication Serial No. 60/019,566 filed on Jun. 11, 1996, each assignedto the assignee of the present application, and further claims thebenefit of U.S. Provisional application Serial No. 60/180,009, filed inthe name of Paul Collins on Feb. 3, 2000, the complete disclosures ofeach of which are incorporated herein by reference.

BACKGROUND OF THE INVENTION

The invention relates to double-ended tuning forks and particularly todouble-ended tuning forks as used in the construction of accelerometerdevices and the cancellation of second order non-linearity thereof.

Non-linearity in accelerometer outputs can lead to significantmeasurement errors in the absence of compensation circuitry. Generally,non-linearity errors occur when inputs are near the full-scale range ofthe instrument or there is vibration along the input axis, butnon-linearity errors may also appear simply because the particularapplication requires an extremely linear response. Instruments usingdouble-ended tuning forks, or DETFs, as inertial reaction force sensorsare particularly vulnerable to errors introduced by non-linearity. Theinherent non-linearity of a force sensor or accelerometer using a singleDETF is typically higher than that of a common high-accuracy, analog,force-rebalance accelerometer, as described in U.S. Pat. Nos. 3,702,073and 4,250,757, for example.

A DETF-based accelerometer, however, possesses real advantages overother accelerometers. For example, a DETF-based accelerometer typicallyprovides smaller size, lower power consumption, and greater ease ofinterface to digital systems. Compensation of DETF-based accelerometernon-linearity provides all these benefits without a serious performancepenalty.

Practical accelerometers in the past have used software compensation ofnon-linearity, or a combination of software and hardware compensation.Software compensation is not viable for other than constant or slowlyvarying acceleration inputs because the processor cannot execute thecompensation commands at frequencies high enough to keep pace with theaccelerometer inputs.

One combined software and hardware compensation approach that has beenused is to infer the input acceleration based on models that depend onthe difference frequency between two DETFs. This approach assumes thatthe DETFs have been designed to possess the same second-ordernon-linearity when subjected to purely axial forces.

The DETFs may be attached either to one or to two independent proofmasses. Dual-proof mass accelerometers are really two separateaccelerometers in the same package. Using dual-proof mass accelerometersleads to difficult matching problems to ensure that the responses of thetwo accelerometers track when the accelerometer experiences vibration orother rapidly-changing inputs.

A common approach to avoiding the common mode tracking problems createdby using two accelerometers in one package is to attach two DETFs to asingle proof mass, arranging them so that displacement of the proof massunder loading simultaneously places one of them in tension and the otherin compression. In practical accelerometers, the exact arrangement ofthe DETFs is dictated by several factors. One factor is the need toincorporate stress isolation, for example, see U.S. Pat. No. 4,766,768,the complete disclosure of which is incorporated herein by reference.Another factor is the necessity of having both DETFs on the same side ofthe proof mass in monolithic silicon accelerometers built with epitaxiallayer DETFs. Other reasons which do not consider the effect of the DETFpositions on the non-linearity of the accelerometer such asmanufacturing tolerances or other processing limitations, or sizerestrictions also dictate the exact arrangement of the DETFs.

General information on the design of vibrating beam accelerometers maybe found in the text by Lawrence entitled Modern Inertial Technology:Navigation, Guidance and Control, Copyright 1993, Springer-Verlag, NewYork.

FIG. 1 shows a plan view of a DETF accelerometer which combines a proofmass 2 and DETFs 4, 6. DETFs 4, 6, however, are positioned at muchdifferent distances 14, 16 from the centerline 8 of the hinges 10, 12suspending proof mass 2. Thus, the respective non-linearity of the twoDETFs do not cancel effectively when the difference frequency is formed,even when the DETFs are designed for the ideal case in whichsecond-order non-linearity, K2, values cancel when subjected to purelyaxial forces. The lack of second-order non-linearity cancellation whenthe difference frequency is formed causes measurement errors and createsdifficulties when DETF force sensors and accelerometers are used inapplications requiring a high degree of linearity.

Above incorporated co-pending parent patent application Ser. No.08/873,048 describes a method for determining relative positioning ofthe DETFs in a dual vibrating beam accelerometer which substantiallyovercome the problems of the prior art by providing positioning of thetwo DETFs which minimizes or substantially eliminates second-order, K2,non-linearity effects. The parent application also provides variousphysical embodiments which place the two DETFs such that the individualDETF second-order values are a minimum and the composite second-orderterms cancel or substantially cancel.

However, as ever greater degrees of linearity are required by more andmore sensitive accelerometer applications, additional fine tuning ofsecond-order, K2, non-linearity effects is required to ensure completeor substantially complete cancellation of the composite second-orderterms of two DETFs in a practical dual vibrating beam accelerometer.

SUMMARY OF THE INVENTION

The present invention overcomes the limitations of the prior art byrecognizing and accounting for the deformation of the DETFs in atwo-DETF, single-proof-mass accelerometer that are not purely axialextensions or compressions, but also involve rotations and transversedisplacements of the ends of the DETFs. The rotations and displacementscreate additional changes in the tine stiffness, beyond those that occurdue to simple stress stiffening effects. The additional stiffnesschanges alter the linearity of the DETFs so that the second-ordereffects such as those due to, for example, Euler buckling loads, do notcancel when the difference frequency is formed.

According to one aspect of the present invention, the present inventionincludes various embodiments which overcome the limitations of the priorart by providing mass balances positioned on each of the two DETFs whichminimize or eliminate second-order, K2, non-linearity effects.

According to another aspect of the present invention, the inventionprovides a double-ended tuning fork (DETF) sensor having first andsecond DETFs, a proof mass, a support frame, and a hinge rotatablysuspending the proof mass from the support frame. The two DETFs arespaced apart and connected between the proof mass and the support frame.The first and second DETFs are each constructed having two tines.According to the invention, mass balances are formed projecting fromeach of the tines of the first DETF and are sized and positioned to forma first second-order non-linearity term associated with the first DETF.Similarly, mass balances projecting from each of the tines of the secondDETF are sized and positioned to form a second second-ordernon-linearity term associated with the second DETF such that the secondsecond-order non-linearity term is substantially equal in sign andmagnitude to the first second-order non-linearity term.

According to one aspect of the invention, the proof mass and supportframe are formed in a silicon wafer having an active epitaxial layerformed on one surface thereof, and each of the DETFs and the massbalances are formed in the active epitaxial layer.

According to another aspect of the invention, the mass balances projectoutwardly from the edges of the tines in a formation substantiallysymmetrical about a longitudinal axis of the respective DETF. Inparticular, the mass balances are formed along an edge of each tine as afunction of the second-order non-linearity term associated with therespective DETF, such that the mass balances adjusts the secondsecond-order non-linearity term associated with each DETF to a valuesubstantially equal in magnitude to the second-order non-linearity termassociated with the other DETF.

According to still another aspect of the invention, the multiple massbalances project from each of the tines.

According to another aspect of the present invention, the inventionprovides methods for sizing and positioning the mass balances on one orboth of the dual-DETFs such that second-order non-linearity of the twoDETFs are equal or substantially equal under the deformations that theyactually undergo in use. Thus, the present invention providescancellation of the composite second-order non-linearity.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a plan view of a double-ended tuning fork device according tothe prior art;

FIG. 2 is a plan view of a double-ended tuning fork device according toan embodiment of co-pending parent patent application Ser. No.08/873,048;

FIG. 3 is an illustration of DETF positioning according to an embodimentof co-pending parent patent application Ser. No. 08/873,048;

FIG. 4 is a flow chart representation describing the iterative methodaccording to one embodiment of co-pending parent patent application Ser.No. 08/873,048;

FIG. 5 is a plan view of a double-ended tuning fork device according toan embodiment of co-pending parent patent application Ser. No.08/873,048;

FIG. 6. is a plan view of a double-ended tuning fork device according toan embodiment of co-pending parent patent application Ser. No.08/873,048;

FIG. 7 is another plan view of a double-ended tuning fork deviceaccording to an embodiment of co-pending parent patent application Ser.No. 08/873,048;

FIG. 8 is a detailed plan view of a double-ended tuning fork deviceaccording to an embodiment of co-pending parent patent application Ser.No. 08/873,048;

FIG. 9 is a detailed plan view of a double-ended tuning fork deviceaccording to an embodiment of the present invention;

FIG. 10 is a detailed plan view of a double-ended tuning fork deviceaccording to an alternative embodiment of the present invention; and

FIG. 11 is a flow chart representation describing the iterative methodaccording to one embodiment of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The mounting positions of the two DETFs on the proof mass and on theouter support structure directly affect second order non-linearityperformance and have resulted in design compromises in the prior art.The invention of the parent application provides various embodimentswhich overcome the problems of the prior art by providing positioning ofthe two DETFs which minimizes or eliminates second-order, K2,non-linearity effects. The parent invention is effective in both high-gforce operation in excess of 1000 g's and in high vibration environmentsas well as in lower force and vibration ranges. Furthermore, the parentinvention is realized in geometry features; thus, no externalenvironment nulls-out or degrades sensor performance. In other words,there is no theoretical limit on the range of g-force input or vibrationenvironment in which the invention may operate.

However, as DETF force sensors and accelerometers are used inapplications requiring ever higher degrees of linearity, further finetuning of second-order non-linearity terms is necessary to eliminatemeasurement errors and ensure cancellation of second-order non-linearityeffects when the difference frequency is formed. The present inventiontherefore provides trim balancing tabs, one or more of which are addedto each DETF to equalize the second-order, K2, non-linearity effects andensure cancellation. Any practical limits on the operational g-forcerange or vibrational environment result from limitations inmanufacturing processes used to practice the invention. The presentinvention provides the overall product goals of excellent performanceand low unit cost.

Various embodiments of the present invention are disclosed. Thepreferred geometry to be used is dictated by the specific designconstraints of a particular application. The present invention providesa method for determining the preferred DETF trim tab configuration anddetermining the fine adjustment and matching of the K2 linearity of thetwo DETFs to achieve the desired composite sensor performance whilerealizing low unit cost.

There are many ways of expressing the non-linearity of the DETFs whichare known to those of skill in the art. The general equations describingDETF performance follow and the terms and coefficients used herein aredefined. Although the calculation of non-linearity can be carried out toan infinite degree or order, typical practical accelerometerapplications carry out the calculations only to the third order becauseof the diminishing effect of higher order non-linearity on DETFperformance. All of the many ways of expressing the third-ordernon-linearity start with a formula associating DETF frequencies, F, withinput acceleration, g. The output frequencies of the two DETFs, F I andF2, are given as: $\begin{matrix}{{F1} = {{\sum\limits_{n = 0}^{3}{a_{n}*g^{n}\quad \text{and}\quad {F2}}} = {\sum\limits_{n = 0}^{3}{b_{n}*g^{n}}}}} & \text{(Eqs.~~1~~and~~2)}\end{matrix}$

where:

an and bn are constants; and

gn=actual input acceleration raised to the nth power.

Thus, the output frequencies of the two DETFs, F1 and F2, at the inputacceleration, g, for n=0, 1, 2, and 3 are given as:

F 1=a 0+(a 1 *g)+(a 2*g ²)+(a 3*g ³)  (Eq. 3)

F 2=b 0 +[b 1*(−g)]+[b 2*(−g ²)]+[b 3*(−g ³)]  (Eq.4)

where:

F1 is the output frequency of the first DETF;

F2 is the output frequency of the second DETF;

g is the actual g-force input sensed by the accelerometer; and

a0 . . . a3 and b0 . . . b3 are constants:

a0, b0 are bias;

a 1, b 1 are scale factor;

a2, b2 are second-order non-linearity; and

a3, b3 are third-order non-linearity.

Since the two DETFs in such an accelerometer flex in opposite directionsfor any given input to the sensor, the sign of the input acceleration,g, of equation (4) is opposite that of equation (3). Cancellation isgiven by:

F 1 −F 2  (Eq. 5)

which expands to: $\begin{matrix}{{{F1} - {F2}} = \frac{\begin{matrix}{{a0} + \left( {{a1}*g} \right) + \left( {{a2}*g^{2}} \right) + \left( {{a3}*g^{3}} \right) -} \\\left\{ {{b0} + \left\lbrack {{b1}*\left( {- g} \right)} \right\rbrack + \left\lbrack {{b2}*\left( {- g^{2}} \right)} \right\rbrack + \left\lbrack {{b3}*\left( {- g^{3}} \right)} \right\rbrack} \right\}\end{matrix}}{\begin{matrix}{\left\lbrack {{a0} - {b0}} \right\rbrack + \left\lbrack {\left( {{a1} + {b1}} \right)*g} \right\rbrack +} \\{\left\lbrack {\left( {{a2} - {b2}} \right)*g^{2}} \right\rbrack + \left\lbrack {\left( {{a3} + {b3}} \right)*g^{3}} \right\rbrack}\end{matrix}}} & \text{(Eq.~~6)}\end{matrix}$

The general form of all the expansions for the indicated g-level, G, is:

G=K 0 +K 1 *g+K 2*g ² +K 3*g ³  (Eq. 7)

where:

G is the indicated acceleration or g-level;

K0 is bias;

K1 is scale factor;

K2 is second-order non-linearity;

K3 is third-order non-linearity; and

g is the actual acceleration or g input level in g's.

Conversion from the individual output frequencies of the two DETFs, F1and F2, to the indicated g-levels, G1 and G2, is achieved by dividingequations (3) and (4), above, by a1 and b1, respectively, such that thescale factor coefficients become equal to 1. Thus, DETF frequency, F1,is converted to indicated g-level, G1, in the form of equation (3)according to: $\begin{matrix}{{G1} = {\frac{F1}{a_{1}} = {\frac{a_{0}}{a_{1}} + \frac{a_{1*g}}{a_{1}} + \frac{a_{2*g^{2}}}{a_{1}} + \frac{a_{3*g^{3}}}{a_{1}}}}} & \text{(Eq.~~8)}\end{matrix}$

which can be rewritten in the form of equation (7) as:

G 1 =K 0 ₁1+*g+K 2 ₁ g+K 3 ₁ *g ³  (Eq. 9)

where: K0 ₁ . . . K3 ₁ are constants associated with the first DETF andare defined as shown in Table 1.

Similarly, DETF frequency, F2 in the form of equation (7), is convertedto indicated g-level, G2, by dividing equation (4) by the constant, b1,to obtain:

G ₁ =K 0 ₁+1*g+K 2 ₁ *g ² +K 3 ₁ *g ³  (Eq. 10)

where: K0 ₁ . . . K3 ₁ are constants associated with the second DETF andare similarly defined as shown in Table 1.

Conversion from the difference frequencies of the two DETFs, F1−F2, tothe indicated g-level, G, is achieved by dividing equation (6), above,by the composite scale factor coefficient,(a1+b1). Thus, the differencefrequency, F1−F2, is converted to indicated g-level, G, in the form ofequation (7). Thus, DETF difference frequency, F1−F2, is converted toindicated g-level, GDETF, in the form of equation (6) according to:$\begin{matrix}{{GDETF} = {\frac{{F1} - {F2}}{\left( {{a1} + {b1}} \right)} = {\frac{\left\lbrack {{a0} - {b0}} \right\rbrack}{\left( {{a1} + {b1}} \right)} + \frac{\left\lbrack {\left( {{a1} + {b1}} \right)*g} \right\rbrack}{\left( {{a1} + {b1}} \right)} + \frac{\left\lbrack {\left( {{a2} - {b2}} \right)*g^{2}} \right\rbrack}{\left( {{a1} + {b1}} \right)} + \frac{\left\lbrack {\left( {{a3} + {b3}} \right)*g^{3}} \right\rbrack}{\left( {{a1} + {b1}} \right)}}}} & \text{(Eq.~~11)}\end{matrix}$

which can be rewritten in the form of equation (7) as:

G=K 0+1 *g+K 2*g ² +K 3*g ³  (Eq. 12)

where:

K0 . . . K3 are constants and are defined as shown in equation (11) andTable 1.

Those of skill in the art will recognize that the actual input level, g,may be estimated from either F1, F2 or F1−F2, so long as the appropriateK values are used to form the correct expression for the associatedindicated output, G1, G2, or GDETF. Table 1 shows the appropriate Kvalues to be used to form the desired expression for the indicatedoutput, G1, G2, or GDETF, used to estimated the actual inputacceleration, g. However, in most DETF applications, measurement of theinput acceleration, g, is based upon the difference frequency, F1−F2,and estimated in terms of GDETF using equations (11) and (12).

TABLE 1 G1 G2 GDETF Parameter, Units Eq. 8 Eq. 10 Eq. 11 Bias, K0, ga0/a1 b0/b1$\frac{\left( {a_{0} - b_{0}} \right)}{\left( {a_{1} + b_{1}} \right)}$

Scale Factor, K1, g/g 1 1 1 Second-order non-linearity, K2, g/g² a2/a1b2/b1$\frac{\left( {a_{2} - b_{2}} \right)}{\left( {a_{1} + b_{1)}} \right.}$

Third-order non-linearity K3, g/g³ a3/a1 b3/b1$\frac{\left( {a_{3} - b_{3}} \right)}{\left( {a_{1} + b_{1)}} \right.}$

Ideally, both DETFs are designed such that a=b2 and thus the compositesecond-order non-linearity term,$\frac{\left( {a_{2} - b_{2}} \right)}{\left( {a_{1} - b_{1}} \right)},$

is zero in ideal conditions when only axial loading exists. In otherwords, the second-order non-linearity terms of the individual DETFs aredesigned to exactly cancel when subjected to purely axial forces.

The coefficients in a practical sensor, however, will not exactly matchdue to design considerations and manufacturing tolerances. Furthermore,in a practical accelerometer, because one end of each of the DETFs isfixed to a solid support structure while the other moves with the proofmass, the two DETFs experience deformations, including rotations andtransverse displacements, not accounted for when the DETFs have beendesigned to have equal second-order non-linearity values when subjectedto purely axial forces. Thus, the second and third order non-linearityvalues, K2 and K3, respectively, will not cancel in a practicalaccelerometer even when the DETFs are designed to be identical.

Various embodiments of the parent invention hereto overcome this lack ofcomposite second-order cancellation by positioning each of the two DETFsin a practical accelerometer which provides substantially completecancellation or elimination of second-order, K2, non-linearity effects.Other aspects of the parent invention provide various physicalembodiments which place the two DETFs such that the individual DETFsecond order values of the two DETFs are a substantially equalized andthe composite second-order terms cancel.

The invention of the parent application provides positioning the ends ofthe DETFs attached to the proof mass, which are the ends that move, suchthat second-order non-linearity of the two DETFs will be equal orsubstantially equal, under the deformations that they actually undergoin use, including rotation and transverse displacement. For example, theK2 second-order non-linearity of the two DETFs will be within about 0 to10 micro-g's of one another. Thus, the second-order non-linearity willbe absent or substantially absent from the difference frequency. Theaccelerometer of the parent application places the ends of the DETFssuch that the second-order terms, a2 and b2, cancel or substantiallycancel when both axial and transverse forces are considered.

With reference to the general expressions of non-linearity above, thecomposite second-order term in the difference frequency cancels when thedifference in the individual DETF coefficient terms, a2−b2, equals zero.Thus, the accelerometer of the parent application places the ends of theDETFs such that a2−b2 is equal to zero, or substantially equal to zero.

FIG. 2 shows a plan view of an accelerometer constructed according tothe parent application. FIG. 2 shows a configuration for the case wherethe two DETFs 20, 22 are the same size and shape and are positioned onopposite sides of the hinge axis 24 of the sensor proof mass 26, wherehinge axis 24 is defined by the centerline of flexures 28, 30. Accordingto the parent invention, positioning of DETFs 20, 22 such that thenumerator, (a2−b2), in the second-order non-linearity equation$\frac{\left( {a_{2} - b_{2}} \right)}{\left( {a_{1} - b_{1}} \right)}$

is zero, or approximately zero, is accomplished by having the movingends of both DETFs 20, 22 at essentially the same distance from hingeaxis 24. In other words, in FIG. 2, distance 32 is equal orapproximately equal to distance 34.

Positioning DETFs 20, 22 within hinges 28, 30 and close to thecenterline 36 of proof mass 26 maintains good common modecharacteristics. For example, when DETFs 20, 22 are misaligned withrespect to centerline 36, a mechanical moment couple may be formedbetween DETFs 20, 22 which could limit the common mode trackingperformance. Common mode tracking performance, or common modecancellation, is the tracking and mutual cancellation of the common moderesponses of two DETFs in a single sensor when the sensor is subjectedto a vibration input or any other rapidly changing input. Such a coupleis avoided according to the embodiment of FIG. 2 when DETFs 20, 22 arealigned with centerline 36.

According to the embodiment of the parent invention shown in FIG. 2,side loading is minimized and good bandwidth is maintained. For example,DETFs 20, 22 operate effectively within a +/−30 percent range offrequency change relative to their nominal no-load operating frequency.DETFs 20, 22 preferably operate within a +/−10 percent range offrequency change. In one example, if the DETFs are designed to operateat a nominal no-load frequency of 100,000 Hz, the change in frequencyover the entire operating range of the sensor, from negative full scaleinput to positive full scale input, is preferably within +/−10 percentof the nominal no-load operating frequency. In the example where theDETFs are designed to operate at a nominal no-load frequency of 100,000Hz, the DETFs are designed to operate in the range of 90,000 Hz to110,000 Hz.

Furthermore, the configuration of FIG. 2 limits unit size and cost whileimproving performance. The performance improvements and reducedsensitivity to thermal effects and external stresses provided by theparent invention reduce sensitivity to manufacturing processes. Thus,lower cost for comparable performance is achieved in a comparable unitsize.

The physical embodiment of FIG. 2 is achieved using the method of theparent invention in which an iterative approach is used to determine thepreferred spacing between DETFs 20, 22 at which the K2 value of thedifference frequency cancels or substantially cancels and the K2 valuesfor individual DETFs 20, 22 are minimized for a given set of designconstraints for a particular application through a sequence ofcalculations not relevant to the present invention but discussed indetail both below and in the parent application.

FIG. 3 is an illustration of DETF 40, 42 positioning for a specificapplication according to one embodiment of the parent invention whichresults in cancellation or substantial cancellation of the compositesecond-order non-linearity, K2, term and in which individual DETFs 40,42 are designed to have minimum second-order non-linearity, K2, values.

In many practical applications the two DETFs are designed to differ inphysical form to maximize performance including, for example, avoidingundesirable interactions between the two DETFs when the frequencyoutputs of the two DETFs cross during transition from positive tonegative input. The cancellation of second order effects can also beaccomplished for DETFs that differ in form. The cancellation requirespositioning the ends at different, but definite, distances from thesensor hinge axis. Each DETF is located at a distance which is adefinite multiple of the length of that DETF.

FIG. 3 shows a configuration for the case where first DETF 40 and secondDETF 42 differ in size and shape and are positioned on opposite sides ofthe hinge axis 44, where hinge axis 44 is defined by the center ofrotation of flexures 46, 48. According to the parent invention,positioning of DETFs 40, 42 such that the numerator, (a2−b2), of thesecond-order non-linearity term,$\frac{\left( {a_{2} - b_{2}} \right)}{\left( {a_{1} - b_{1}} \right)},$

is zero, or approximately zero, is accomplished by having the moving endof first DETF 40 positioned at a first distance 52 from hinge axis 44and the moving end of second DETF 42 positioned at a second distance 54from hinge axis 44. For example, the particular application described inFIG. 3, DETFs 40, 42 are 1864 microns and 1851 microns in length,respectively, formed in a wafer having a standard thickness of 525microns. The positioning of DETFs 40, 42 according to the parentinvention which results in substantially complete or substantiallycomplete cancellation of the composite second-order non-linearity, K2,terms is accomplished by positioning first DETF 40 at a distance 52equal to 2.44 times the length of DETF 40 from hinge axis 44 andpositioning second DETF 42 at a distance 54 equal to 2.61 times thelength of DETF 42 from hinge axis 44. In other words, in FIG. 3,distance 52 is equal to 2.44 times the length of first DETF 40 anddistance 54 is equal to 2.61 times the length of second DETF 42.Positioning DETFs 40, 42 within hinges 48, 50 and close to thecenterline 56 of proof mass 46 maintains good common modecharacteristics as discussed in reference to the embodiment of FIG. 2,above. Also as discussed in reference to FIG. 2, above, according to theembodiment shown in FIG. 3, side loading is minimized and good bandwidthis maintained.

Additional embodiments of the parent invention in the form of FIG. 3 aredetailed in Table 2 including positioning of the two DETFs relative tohinge axis 44 at distances 52, 54 which are multiples of the individuallengths of each DETF 40, 42, and the degree of cancellation or compositesecond-order non-linearity term, K2, in hertz. The embodiment of FIG. 3may utilize DETFs according to co-pending U.S. patent application Ser.No. 08/873,048 filed Jun. 11, 1997, the complete disclosure of which isincorporated herein by reference, which is similarly assigned to theassignee of the present patent application. Optionally, the DETFs may beconstructed using any of the designs known to those of skill in the art.

The additional embodiments of FIG. 3 described in Table 2 may alsoutilize DETFs according to co-pending U.S. patent application Ser. No.08/873,048 filed Jun. 11, 1997.

TABLE 2 Distance 52 Distance 54 FIG. 3 (multiple of (multiple ofComposite Embodiments DETF 1 length) DETF 2 length) K2 (Hz) 1 (shown)2.44 2.61 0 2 1.55 1.91 0 3 2.00 2.00 −70 4 2.56 2.56 −5 5 3.00 3.00 −4

FIG. 4 is a flow chart representation describing the iterative methodused to achieve the physical embodiments shown in FIGS. 2 and 3. Onemethod for implementing the parent invention is through the sequence ofcalculations outlined below, which are greatly simplified by the factthat none of the finite element model changes made in the course of thecalculations cause a significant change in the accelerometer scalefactor, where scale factor, K1, is the sensor's sensitivity to inputforce or acceleration. According to one embodiment of the presentinvention, the accelerometer of the parent invention may, for example,be designed as follows:

First step 60: DETF design. Design DETFs 20, 22 using classical formulaeor finite element methods to give the desired nominal no-loadfrequencies and scale factors. Scale factor may be either frequencychange per unit load or frequency change per unit extension. Use thewell-known design rules, for example, those found in Lawrence's ModemInertial Technology: Navigation, Guidance and Control, to ensure thatDETFs 20, 22 possess equal, or very nearly equal, second ordernon-linearity, using the desired unit of measure, for example,micro-g/g2, milli-g/g2, or hertz. According to one embodiment of theparent invention, DETFs 20, 22 are designed using classical formulae orfinite element methods to have second order non-linearity which areeither a minimum or zero. Classical formulae for designing DETFs to givethe desired nominal no-load frequencies and scale factors are describedin, for example, U.S. Pat. No. 4,372,173, the complete disclosure ofwhich is incorporated herein by reference. The finite element models maybe created using, for example, ANSYS, NASTRAN, COSMOS, or other suitablefinite element modeling programs capable of Eigen value extraction.

Second step 62: Finite element model creation. Create a finite-elementmodel including, as a minimum, DETFs 20, 22 and sensor hinges 28, 30connected together at one end by a very stiff structure, preferably amassless structure, representing the proof mass. The DETF length istypically on the order of 1,500 microns. Hinges 28, 30 are preferably onthe order of 100 to 200 microns in length, and are preferably as thickas possible without seriously degrading sensor performance. Designfactors beyond the scope of this or the parent specification may enterin the selection of hinge 28, 30 dimensions. Generally, the ratio ofDETF length to hinge length is preferably as large as practical, forexample, the ratio of DETF length to hinge length is preferably in arange from 8:1 to 20:1 or more. The other ends of the DETFs are fixed orsolidly connected to an immobile structure. The distance from the planedefined by DETFs 20, 22 to hinge axis 24 should be the same as thatintended for use in the actual accelerometer. In a typical siliconsensor, this distance is a fixed fraction of the wafer thickness.Typically, the hinge is formed at the centerline of the substrate suchthat the distance from the plane defined by the DETFs to the hinge axisis one half the thickness of the substrate.

Third step 64: Application of full-scale loading. Within the finiteelement model, apply model-forces to the connecting structure to causeit to rotate to the degree the actual proof mass 26 is expected to turnfor a full-scale input. In addition, apply full-scale accelerationloading directly to DETFs 20, 22, so that the finite element modelresults includes the effects of DETF 20, 22 deformations due to the sideloading that will be present in an actual accelerometer. Attainment of afull-scale displacement can be deduced from the frequency changes inDETFs 20, 22 compared to their unloaded values.

Fourth step 66: Model analysis. Use the non-linear analysis capabilitiesof the finite element analysis program to find the vibration frequenciesof DETFs 20, 22 in a deformed, pre-stressed state. Record thefrequencies.

Fifth step 68: Repetitive vibration frequency analysis. Repeat vibrationfrequency analysis of fourth step 66 to find the frequencies of DETFs20, 22 for at least four more loads between negative full-scale andpositive full-scale. Use curve-fitting techniques known to those ofskill in the art to find the non-linearity in the difference frequency.

Sixth step 70: Repetitive finite element analysis. Create finite-elementmodels for various DETF-to-DETF spacings between DETFs 20, 22. Repeatloading and vibration frequency analysis of third step 64 through fifthstep 68 for various DETF-to-DETF spacings between DETFs 20, 22 to createa model of second-order non-linearity as a function of DETF-to-DETFspacing.

Seventh step 72: DETF-to-DETF positioning/spacing selection. Select thepositions of DETFs 20, 22, or the spacing between DETFs 20, 22 where thecomposite second-order non-linearity is either a minimum or zero. Inother words, select the positions of DETFs 20, 22, or the spacingbetween DETFs 20, 22 where the respective second-order non-linearityvalues of DETFs 20, 22 cancel or substantially cancel.

Eighth step 74: Validation of selected DETF-to-DETF positioning/spacingselection. Create a complete design embodying the DETF-to-DETFpositioning or spacing selected in seventh step 72, being certain thatthe center of mass of the proof mass is located appropriately to producethe rotations at full-scale that were assumed in defining DETF 20,22positions.

Those of skill in the art will realize that the optimal design from thestandpoint of K2 cancellation may not be the best from otherstandpoints. In particular, implementation of K2 cancellation mayrequire a larger piece of silicon to manufacture. Those of skill in theart will realize that selection of the best DETF positions according tothe parent invention for the overall accelerometer should consider allpertinent factors, not just K2. Pertinent factors may include, forexample, the desired scale factor, third-order affects, and otherperformance goals of the sensor or accelerometer.

FIG. 5 illustrates another embodiment of the parent invention. Theembodiment of FIG. 5 includes a projection 80 on the proof mass 82 whichallows positioning the DETFs 84, 86 within the hinges 88, 90 and closeto the centerline 92 of proof mass 82 between hinges 88, 90. As noted inthe discussion of FIG. 2, above, positioning DETFs 84, 86 away fromcenterline 92 between hinges 88, 90 may reduce common mode trackingperformance. However, according to the embodiment of the inventiondepicted in FIG. 5, DETFs 84, 86 are moved close to centerline 92 whichlimits the effects of geometry on common mode tracking performance.DETFs 84, 86 may be positioned as close to centerline 92 as processingtechniques allow, but DETFs 84, 86 are preferably separated by a minimumdistance such that cross-coupling is avoided. In one example, DETFs 84,86 are separated by 800 microns.

FIG. 6 illustrates another embodiment of the parent invention. Theembodiment of FIG. 6 includes projections 100, 102 on the sides of proofmass 104 which allow positioning the DETFs 106, 108 on either side ofproof mass 104 and outside hinges 110, 112 at distances 114, 116 fromcenterline 118. As noted in the discussion of FIG. 2, above and in theparent application, positioning DETFs 106, 108 away from centerline 118may reduce common mode tracking performance. However, according to theembodiment of the parent invention depicted in FIG. 6, the relativelygreater cross-axis stiffness of hinges 110, 112 compared to thenegligible stiffness of DETFs 106, 108 limits the effects of geometry oncommon mode tracking performance. In a practical accelerometer,cross-axis stiffness of hinges 110, 112 may be 100 or more times greaterthan the stiffness of DETFs 106, 108. Further, the embodiment of FIG. 6provides the minimum sensor area for a desired sensor response or scalefactor, which increases the number of mechanisms that can be fabricatedper silicon wafer, thereby lowering unit cost. According to oneparticular embodiment of the parent invention according to FIG. 6, DETFs106, 108 are 1804 microns and 1800 microns in length, respectively,formed in a 525 micron thick wafer. The moving end of DETF 106 ispositioned at 1.47 times the length of DETF 106 from the hinge axis 119formed at the center of rotation of hinges 110, 112 and the moving endof DETF 108 is positioned at 0.32 times the length of DETF 108 fromhinge axis 119.

FIG. 7 illustrates another embodiment of the parent invention. Theembodiment of FIG. 7 eliminates the projection on the proof mass shownin FIGS. 2, 3 and 5 by mounting the DETFs 120, 122 to a frame projection124 within the proof mass 126 structure. The embodiment of FIG. 7includes a new configuration for proof mass 126. The embodiment of FIG.7 maximizes pendulousity for proof mass size, minimizes hinge sideloading which may reduce common mode tracking performance and allowsmaximum separation between flexure hinges. pendulousity of the proofmass is maximized by positioning DETFs 120, 122 within the hinges 128,130 and close to the centerline 132 of proof mass 126 to maintain goodcommon mode performance. Thus, side loading is reduced and goodbandwidth as defined above is maintained. Unit size and cost are kept toa minimum while performance is improved. By eliminating projection 80and thus reducing the overall area of the mechanism, the embodiment ofFIG. 7 also increases the number of mechanisms that can be fabricatedper silicon wafer, thus, lowering unit cost.

The tines of DETFs are formed in the silicon wafer by an etching processwell known to those of skill in the art. The tines of DETFs aretypically formed in the silicon wafer by masking the wafer with a maskhaving the desired tine shape and coating the exposed areas of the waferwith a substance which is impervious to silicon-etching chemicals. Themask is removed and the wafer is exposed to a silicon-etching chemicalwhereby the wafer material around the DETF tines is dissolved thusforming the tines in the silicon wafer. The wafer may be repeatedlyexposed to the masking and etching process using different shaped masksdesigned to progressively reveal a tine having the desired shape and thedesired degree of detail.

FIG. 8 is a detailed plan view of a double-ended tuning fork deviceaccording to an embodiment of the parent invention. The structure ofFIG. 7 including DETFs 120, 122 is formed in part using silicon etchingtechniques known to those of skill in the art and described, forexample, in U.S. Pat. Nos. 4,597,003 and 4,783,237 and co-pending patentapplication Ser. No. 09/350,323, entitled Method of Machining Glass,filed in the name of Amy V. Skrobis on Jul. 9, 1999, the completedisclosures of which are incorporated herein by reference. FIG. 8 showsan enlarged view of DETF 120. The effective interface points betweenDETF 120 and proof mass 126 and between DETF 120 and projection 124 arethe effective end points of the DETF tines. Possible adjustments of theinterface to both proof mass 126 and frame projection 124 are indicatedin FIG. 8 by dotted lines. A single repetition of the masking andetching process can relocate the effective interface point as indicatedby arrows 134, 136. Relocation of the effective interface point is asimple cost effective approach for tuning both individual DETFs 120, 122second-order non-linearity, K2, terms and the composite second-ordernon-linearity, K2, term. This adjustment or tuning technique used inconjunction with any of the various embodiments described hereinprovides additional cost effective and schedule effective performanceoptimization of vibrating beam force sensors and accelerometers usingonly one mask for the relocation of the effective interface point.

Preferred embodiments of the parent invention have been described.Variations and modifications will be readily apparent to those of skillin the art.

The Present Invention

In a practical sensor, the second-order non-linearity terms of theindividual DETFs, a2 and b2, will not exactly match because of designconsiderations and manufacturing tolerances. Furthermore, in a practicalaccelerometer, because one end of each of the DETFs is fixed to a solidsupport structure while the other moves with the proof mass, the twoDETFs experience deformations, including rotations and transversedisplacements, not accounted for when the DETFs have been designed tohave equal second-order non-linearity values when subjected to purelyaxial forces. Thus, the second and third order non-linearity values, K2and K3, respectively, will not cancel in a practical accelerometer evenwhen the DETFs are designed to be identical.

Various embodiments of the present invention overcome this lack ofcomposite second-order cancellation by providing one or more trimbalancing tabs positioned at predetermined points along the length ofeach of the two DETFs in a practical accelerometer which providessubstantially complete cancellation or elimination of second-order, K2,non-linearity effects. Other aspects of the present invention providevarious physical embodiments which place the trim balancing tabs or“trim tabs,” on the two DETFs such that the individual DETF second ordervalues of the two DETFs are substantially equalized and the compositesecond-order terms cancel.

A double ended tuning fork (DETF) resonator has an inherent non-linearscale factor term, known as K2. One type of DETF is described in US Pat.No. 5,996,411. This term is trimmed or “fine-tuned” on individual DETFsby the introduction of a mass balance located on the edge of the DETFtines. Depending upon the mass volume and location of the balance on thetines, the second-order, K2, term can be adjusted higher or lower. Whentwo DETF are employed in a sensor, such as an accelerometer, the K2 ofeach DETF can be adjusted to be of the same sign and magnitude. Theresult is a highly linear practical accelerometer.

The present invention includes positioning trim balancing tabsprojecting from each of the DETFs such that second-order non-linearityof the two DETFs are of substantially identical sign and magnitude underthe deformations that they actually undergo in use, including rotationand transverse displacement. Thus, the second-order non-linearity isthus absent by cancellation from the difference frequency. Theaccelerometer of the present invention places the trim balancing tabs onthe DETFs such that the second-order terms, a2 and b2, cancel orsubstantially cancel when both axial and transverse forces areconsidered.

As noted above, with reference to the general expressions ofnon-linearity above, the composite second-order term in the differencefrequency cancels when the difference in the individual DETF coefficientterms, a2−b2, equals zero. Thus, the accelerometer of the presentinvention places the trim tabs on the DETFs such that the expressiona2−b2 is equal to zero, or approximately zero.

The attachment locations of the trim tabs on the two DETFs directlyaffect second order non-linearity performance. The invention of thepresent application provides various embodiments which overcome theproblems of the prior art by providing trim tabs positioned along thelength of each of the two DETFs which minimize or eliminatesecond-order, K2, non-linearity effects by equalizing the individualDETF second-order, K2, non-linearity coefficient terms. The presentinvention is effective in both high-g force operation in excess of 100g's and in high vibration environments as well as in lower force andvibration ranges. Furthermore, the present invention is realized ingeometry features; thus, no external environment nulls-out or degradessensor performance. In other words, there is no theoretical limit on therange of g-force input or vibration environment in which the inventionmay operate.

The tines of DETFs, including the trim balancing tabs, are formed in thesilicon wafer by an etching process well known to those of skill in theart, such as one of those processes mentioned above. The tines of DETFsare typically formed in the silicon wafer by masking the wafer with amask having the desired tine shape and coating the exposed areas of thewafer with a substance which is impervious to silicon-etching chemicals.The mask is removed and the wafer is exposed to a silicon-etchingchemical whereby the wafer material around the DETF tines is dissolvedthus forming the tines in the silicon wafer. The wafer may be repeatedlyexposed to the masking and etching process using different shaped masksdesigned to progressively reveal a tine having the desired shape and thedesired degree of detail, including the trim balancing tabs.

FIG. 9 is a detailed plan view of a double-ended tuning fork deviceaccording to an embodiment of the present invention. The structure ofFIG. 7 including DETFs 120, 122 is formed in part using silicon etchingtechniques known to those of skill in the art and described, forexample, in above incorporated U.S. Pat. Nos. 4,597,003 and 4,783,237and co-pending patent application Ser. No. 09/350,323. FIG. 9 shows anenlarged view of DETF 120 configured with trim balancing tabs accordingto the present invention. One or more mass balances or “trim balancingtabs” 200 are located on the edge of the DETF tines 202A and 202B fortrimming or “fine-tuning” the second-order, K2, term on individualDETFs. The mass volume and location of trim balance tabs 200A and 200Bon respective tines 202A, 202B is adjusted according to the invention toadjust the second-order, K2, higher or lower. Preferably, trim tabs 200are located along the length of tines 202 between the electrostaticdrive combs 204 and the effective end points 206 of the DETF tines.

Preferred adjustments of the location of trim tabs 200 are indicated inFIG. 9 by arrows 208. Additionally, the mass volume of each trim tab 200is individually adjusted to match or equalize the second-order, K2,non-linearity terms such that the respective non-linearity of the twoDETFs effectively cancel when the difference frequency is formed.Introduction of the trim balance tabs 200 on the tines of DETF 120 is asimple, cost effective approach for tuning both individual DETFs 120,122 second-order non-linearity, K2, terms and the composite second-ordernon-linearity, K2, term. This adjustment or tuning technique used inconjunction with any of the various embodiments described hereinprovides additional cost effective and schedule effective performanceoptimization of vibrating beam force sensors and accelerometers usingonly one mask for the relocation of the effective interface point.

FIG. 10 illustrates another embodiment of the trim balancing tabs of theinvention in which multiple trim balancing tabs 200 through 200N areintroduced along the length of each tine 202 of DETF 20. Trim balancingtabs 200 through 200N of an appropriate mass volume are moved asindicated by arrows 208 to appropriate locations along the length ofeach tine of DETF 20 as required to match or equalize the second-order,K2, non-linearity terms of the two DETFs. Trim balancing tabconfigurations, i.e., mass volumes, quantities and locations, aredetermined for each application as a function of the desired degree oflinearity in combination with other design considerations andconstraints.

FIG. 11 is an illustrative flow chart representation 300 describing theiterative method used to achieve the physical embodiments of a DETFaccelerometer shown in FIGS. 2 and 3, or another of the physicalembodiments described herein, using the physical embodiments of a DETFshown in FIGS. 9 and 10. One method for implementing the parentinvention is through the sequence of calculations outlined below, whichare greatly simplified by the fact that none of the finite element modelchanges made in the course of the calculations cause a significantchange in the accelerometer scale factor, where scale factor, K1, is thesensor's sensitivity to input force or acceleration. According to oneembodiment of the present invention, the accelerometer of the parentinvention may, for example, be designed as follows:

First step 310: DETF design. Design DETFs 20, 22 using classicalformulae or finite element methods, as described above in step 60 ofFIG. 4, to give the desired nominal no-load frequencies and scalefactors. As mentioned above, scale factor may be either frequency changeper unit load or frequency change per unit extension. The well-knowndesign rules discussed above are used to ensure that DETFs 20, 22possess equal, or very nearly equal, second order non-linearity, usingthe desired unit of measure, for example, micro-g/g2, milli-g/g2, orhertz. According to one embodiment of the present invention, DETFs 20,22 are designed to have second order non-linearity which are either aminimum or zero. DETFs 20, 22 are designed using classical formulae asdescribed in, for example, above incorporated U.S. Pat. No. 4,372,173,or using finite element methods, the finite element models may becreated using, for example, ANSYS, NASTRAN, COSMOS, or other suitablefinite element modeling programs capable of Eigen value extraction.

Second step 320: Finite element model creation. Create a finite-elementmodel, as described above in step 62 of FIG. 4, including, as a minimum,DETFs 20, 22 and sensor hinges 28, 30 connected together at one end by avery stiff structure, preferably a massless structure, representing theproof mass. The DETF length is typically on the order of 1,500 microns.Hinges 28, 30 are preferably on the order of 100 to 200 microns inlength, and are preferably as thick as possible without seriouslydegrading sensor performance. Design factors beyond the scope of this orthe parent specification may enter in the selection of hinge 28, 30dimensions. Again, the ratio of DETF length to hinge length is generallypreferably as large as practical, for example, the ratio of DETF lengthto hinge length is preferably in a range from 8:1 to 20:1 or more. Theother ends of the DETFs are fixed or solidly connected to an immobilestructure. The distance from the plane defined by DETFs 20, 22 to hingeaxis 24 is preferably the same as that intended for use in the actualaccelerometer. As discussed above, in a typical silicon sensor, thisdistance is a fixed fraction of the wafer thickness. Typically, thehinge is formed at the centerline of the substrate such that thedistance from the plane defined by the DETFs to the hinge axis is onehalf the thickness of the substrate.

Third step 330: Application of full-scale loading. As described above instep 64 of FIG. 4, within the finite element model, apply model-forcesto the connecting structure to cause it to rotate to the degree theactual proof mass 26 is expected to turn for a full-scale input. Inaddition, apply full-scale acceleration loading directly to DETFs 20,22, so that the finite element model results includes the effects ofDETF 20, 22 deformations due to the side loading that will be present inan actual accelerometer. Attainment of a full-scale displacement isdeduced from the frequency changes in DETFs 20, 22 compared to theirunloaded values.

Fourth step 340: Model analysis. Use the non-linear analysiscapabilities of the finite element analysis program to find thevibration frequencies of DETFs 20, 22 in a deformed, pre-stressed state,as described above in step 66 of FIG. 4. Record the frequencies.

Fifth step 350: Repetitive vibration frequency analysis. Repeatvibration frequency analysis of fourth step 340 to find the frequenciesof DETFs 20, 22 for multiple, preferably at least four, more loadsbetween negative full-scale and positive full-scale, as described abovein step 68 of FIG. 4. Use curve-fitting techniques known to those ofskill in the art to find the non-linearity in the difference frequency.

Sixth step 360: Repetitive finite element analysis. Createfinite-element models for DETFs 20,22 having various trim balance tabs200. In other words, fit the finite-element models for DETFs 20, 22 withtrim balance tabs 200 having various mass volumes and locations alongthe length of the tines. As mentioned above, trim tabs 200 arepreferably located along the length of tines 202 between the drive combs204 and the effective end points 206 of the DETF tines. The comb tootharea is thereby effectively reserved for the comb drive. Variousalternative convenient positions for trim tabs 200 are also contemplatedby the present invention. For example, trim tabs 200 are alternativelylocated within the field of the comb drive, either as a functional or anonfunctional comb tooth 204. Whether functional or nonfunctional, trimtab 200 fashioned as a comb tooth 204 within the comb drive field isalternatively either sized similarly to comb teeth 204, or is sizedeither larger or smaller than comb teeth 204, as recommended by therepetitive finite element analysis.

Typically, the second-order, K2, non-linearity term for a single DETF ison the order of a few hundred micro-g's to 500 micro-g's or more.However, the invention of the parent application provides positioningthe ends of the DETFs attached to the proof mass, which are the endsthat move, such that second-order, K2, non-linearity of the two DETFswill be the same or substantially the same, under the deformations thatthey actually undergo in use, including rotation and transversedisplacement. The accelerometer of the parent application places theends of DETFs 20, 22 such that the second-order terms, a2 and b2, cancelor substantially cancel when both axial and transverse forces areconsidered. For example, the difference in the second-order, K2,non-linearity of the two DETFs is often reduced into the range of only afew micro-g's, even within a range of about 0 to 10 micro-g's. Thus, thesecond-order non-linearity is absent or substantially absent from thedifference frequency before fine tuning the respective second-order, K2,non-linearity terms according to the present invention. The presentinvention is most effective in combination with configurations of DETFshaving common mode differences in this narrow 0 to 10 micro-g range.

With reference to the general expressions of non-linearity above, thecomposite second-order term in the difference frequency cancels when thedifference in the individual DETF coefficient terms, a2−b2, equals zero.Thus, while the accelerometer of the parent application places the endsof the DETFs such that a2−b2 is small or nearly equal to zero, theaccelerometer of the present invention adds one or more trim balancingtabs 200 through 200N that substantially equalize the coefficient terms,a2, b2, and drive the composite second-order term in the differencefrequency substantially to zero.

According to the iterative method of the invention for determining theconfiguration of trim balancing tabs 200, a slight increase or decreasein the second-order, K2, non-linearity term for a single DETF is gainedby moving trim tabs along the length of tines 202. Increasing the massvolume of trim tabs 200 and moving them toward the center or comb driveportion of tines 202 generally increases the effect on the second-order,K2, non-linearity term, while decreasing the mass volume and moving trimtabs 200 toward the effective end points 206 of DETF 20 has less effecton the second-order, K2, non-linearity term. Trim tabs 200 are generallyexpected to have substantially the same configuration on each tine 202of DETF 20, i.e., the trim tabs 200 are symmetric along the longitudinalaxis of DETF 20. Trim tabs 200 are preferably fabricated in the plane ofDETF 20 to avoid changing the thickness of the active layer, therebymaintaining manufacturability.

Loading and vibration frequency analysis of third step 330 through fifthstep 350 are repeated, as described above in step 70 of FIG. 4, forvarious trim tab 200 configurations, i.e., mass volumes, quantities andlocations, to create a model of second-order non-linearity as a functionof trim tab configuration.

Seventh step 370: DETF trim tab configuration selection. Select the massvolume, quantity and location or locations of trim balance tabs 200 forDETFs 20, 22 where the composite second-order non-linearity is eitheridentical or substantially identical for each of DETFs 20, 22. In otherwords, select the mass volume and positions of trim balance tabs 200 forDETFs 20, 22 where the respective second-order non-linearity values ofDETFs 20, 22 cancel or substantially cancel.

Eighth step 380: Validation of selected DETF trim balancing tabconfiguration selection. As described above in step 74 of FIG. 4, createa complete design embodying the DETF trim balancing tab configurationselected in seventh step 370, being certain that the center of mass ofthe proof mass is located appropriately to produce the rotations atfull-scale that were assumed in defining DETF 20, 22 positions.

Those of skill in the art will realize that the optimal design from thestandpoint of K2 cancellation may not be the best from otherstandpoints. In particular, implementation of K2 cancellation mayrequire a larger piece of silicon to manufacture. Those of skill in theart will realize that selection of the best DETF positions according tothe parent invention and trim balancing tab configuration according tothe present invention for the overall accelerometer should consider allpertinent factors, not just K2. Pertinent factors may include, forexample, the desired scale factor, third-order affects, and otherperformance goals of the sensor or accelerometer.

While the preferred embodiment of the invention has been illustrated anddescribed, it will be appreciated that various changes can be madetherein without departing from the spirit and scope of the invention.

What is claimed is:
 1. A double-ended tuning fork (DETF) sensorcomprising: a first and a second DETF, each of said DETFs having firstand second tines joined at first and second ends; a proof mass; asupport frame; at least one hinge rotatably connecting said proof massto said support frame; said first ends of said DETFs spaced apart andconnected to said proof mass and said second ends of said DETFsconnected directly to said support frame; and a mass balance disposed oneach tine of said first DETF such that a first second-ordernon-linearity term associated with said first DETF is substantiallyequal in magnitude to a second second-order non-linearity termassociated with said second DETF.
 2. The DETF sensor as recited in claim1, wherein each of said DETFs and said mass balances are formed in anactive layer formed on a substrate.
 3. The DETF sensor as recited inclaim 2, wherein said mass balances are formed substantiallysymmetrically about a longitudinal axis of said first DETF.
 4. The DETFsensor as recited in claim 3, wherein each of said mass balances areformed along the same edge of each said tine of said first DETF.
 5. TheDETF sensor as recited in claim 4, wherein said mass balances arepositioned along an edge of each said tine of said first DETF as afunction of said second-order non-linearity term associated with saidfirst DETF.
 6. The DETF sensor as recited in claim 5, wherein the massbalances further comprise mass balances projecting from each said tineof said first DETF.
 7. The DETF sensor as recited in claim 6, whereinthe mass balances further comprise a plurality of mass balancesprojecting from each said tine of said first DETF.
 8. The DETF sensor asrecited in claim 7, further comprising a mass balance positioned on eachtine of said second DETF and adjusting said second second-ordernon-linearity term associated with said second DETF to a valuesubstantially equal in magnitude to said first second-ordernon-linearity term associated with said first DETF.
 9. A double-endedtuning fork (DETF) sensor, comprising: a first and a second DETF, eachof said DETFs having a first end and a second end; a proof mass; asupport frame, at least one hinge rotatably connecting said proof massto said support frame; said first ends of said DETFs spaced apart andconnected to said proof mass and said second ends of said DETFsconnected to said support frame; wherein said first DETF and said secondDETF are constructed having two tines; first mass balances projectingfrom each of said tines of said first DETF, said first mass balancessized and positioned to form a first second-order non-linearity termassociated with said first DETF; and second mass balances projectingfrom each of said tines of said second DETF, said second mass balancessized and positioned to form a second second-order non-linearity termassociated with said second DETF such that said second second-ordernon-linearity term is substantially equal in sign and magnitude to saidfirst second-order non-linearity term.
 10. The DETF sensor as recited inclaim 9, wherein: said proof mass and said support frame are formed in asilicon wafer having an epitaxial layer formed on one surface thereof;and each of said DETFs and said first and second mass balances areformed in said epitaxial layer.
 11. The DETF sensor as recited in claim10, wherein at least one of said first and second mass balances furthercomprises a plurality of mass balances.
 12. A double-ended tuning fork(DETF) sensor, comprising: a generally planar silicon substrate formedwith an epitaxial layer on one surface thereof; first and second DETFsformed in said epitaxial layer, each of said DETFs having a first andsecond tine joined at a first end and a second end; a support frameformed in said substrate, a proof mass formed in said substrate, saidproof mass rotatably suspended from said support frame; said first endsof said DETFs spaced apart and connected to said proof mass and saidsecond ends of said DETFs connected to said support frame; first trimbalancing tabs projecting outwardly from each said tine of said firstDETF, said first trim balancing tabs sized and positioned to form afirst second-order non-linearity term associated with said first DETF;second trim balancing tabs projecting outwardly from each said tine ofsaid second DETF, said second trim balancing tabs sized and positionedto form a second second-order non-linearity term associated with saidsecond DETF, and wherein said size and said position of each of saidfirst trim balancing tabs and said second trim balancing tabs aredetermined such that said first and said second second-ordernon-linearity terms are substantially equalized in sign and magnitude.13. The DETF sensor as recited in claim 12, wherein said first andsecond trim balancing tabs are sized and positioned as a function ofsecond-order non-linearity resulting from deformations normallyexperienced in use.
 14. The DETF sensor as recited in claim 13, whereinsaid deformations further comprise rotations and transversedisplacements experienced in actual use.